The Southern Health Board (SHB) is investigating more than 20 complaints from members of the public in relation to the smoking ban.
A spokeswoman for the SHB, which covers the Cork and Kerry region, said yesterday that the 20 complaints came to the authorities from a free phone line operated by the Office of Tobacco Control.
The SHB's 14 environmental health officers are now to investigate the complaints to determine if a prosecution is possible.
"These complaints have come from across the counties. As regards their breakdown, they could be in relation to 20 pubs but could also be from restaurants or factories.
"As yet we have had no convictions in the SHB area, but these complaints will be fully investigated.
"The implementation process is in its earliest stages at present, and really is only beginning," said the spokeswoman.
The health board added that while there was a schedule of planned inspections, the complaints made would receive priority.
Gardaí in the Cork area have also received complaints, especially against publicans, and have referred them to the Office of Tobacco Control, which in turn contacts the relevant health board.
The SHB spokeswoman added that the emphasis in the initial period of enforcement was to liaise with employers, and to support them in their implementation of the ban.
"Initial visits will focus on compliance-building, provision of advice and information, and assessment of the level of commitment demonstrated by the proprietor or service in enforcing the legislation.
"There is a high degree of compliance within the SHB area - it is going very smoothly," she added.
A spokesman for the Vintners' Federation of Ireland in the southern region said last night that there was a high level of compliance by publicans.
A spokesman for the Office of Tobacco Control, which is compiling details of the complaints received on the phone line, said he could not provide figures for other parts of the State.
The body would, however, issue a report "in a few weeks time".